1. Multiplying Uniformly Continuous functions

If f and g are two uniformly continuous functions, I need to show that their product is also uniformly continuous. I started with the definition of uniform continuity but got nowhere...

2. You're not going to get anywhere, either, unless you restrict the domain. Is $\displaystyle x\mapsto x$ uniformly continuous on $\displaystyle \mathbb{R}$? What about $\displaystyle x\mapsto x^2$?

3. Sorry I'm not sure what you mean by that...

4. Let $\displaystyle f:\mathbb{R}\to\mathbb{R}$ be given by $\displaystyle f(x)=x$. Is f uniformly continuous? What about the product of f with f?

5. yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one

6. $\displaystyle f(x)=x$ is trivially uniformly continuous since $\displaystyle \left| f(x)-f(y) \right|=\left| x-y \right|,$ so the product is not uniformly continuous, namely $\displaystyle f(x)f(x)=x^2$ is not uniformly continuous.

7. lol but this is isn't making sense since I'm supposed to show the product IS uniformly continuous

8. IT'S NOT TRUE, and i've provided a counterexample.

9. lol you're right, now what if both functions were bounded?

10. Originally Posted by CrazyCat87
yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one
I accept with information: is trivially uniformly continuous since
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